A257522 -id:A257522 - cp's OEIS frontend

A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

0, 16, 9, 0, 20, 12, 441, 64, 16, 90, 1089, 36, 4212, 98, 225, 0, 272, 144, 549081, 25, 567, 2156, 13225, 48, 144, 650, 81, 98, 142100, 150, 71622369, 256, 363, 578, 1225, 64, 1332, 684, 468, 360, 41984, 252, 521345889, 198, 180, 559682, 108241, 144, 63, 400, 127449, 117, 1755572, 108, 2420, 392, 4275, 568458

Offset: 1

Felix Huber, Feb 07 2025

nonn

a(1) = a(4) = a(16) = 0. Proof: See Huber link.

k > n exists for n > 16.

			a(3) = 9 because sqrt(1/3 + 1/4) = sqrt(7/12) is irrational, sqrt(1/3 + 1/5) = sqrt(8/15) is irrational, sqrt(1/3 + 1/6) = sqrt(1/2) is irrational, sqrt(1/3 + 1/7) = sqrt(10/21) is irrational, sqrt(1/3 + 1/8) = sqrt(11/24) is irrational, but sqrt(1/3 + 1/9) = sqrt(4/9) = 2/3 is rational.

Felix Huber, Proof that a(1) = a(4) = a(16) = 0

Cf. A002350, A024352, A076600, A378501, A379816.

Cf. A357372, A257522.

A379815:=proc(n)
    local k;
    if n=1 or n=4 or n=16 then
        return 0
    else
        for k from n+1 do
            if type(sqrt(1/n+1/k),rational) then
                return k
            fi
        od
    fi;
end proc;
seq(A379815(n),n=1..58);

a(n) = if ((n==1) || (n==4) || (n==16), return(0)); my(k=n+1); while (!issquare(1/n + 1/k), k++); k; \\ Michel Marcus, Feb 08 2025

a(n) <= n*A002350(n)^2 - n if n is not a square; a(m^2) <= A076600(m)^2. - Jinyuan Wang, Feb 11 2025

A357372 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the numerator of 1/n + 1/k.

Original entry on oeis.org

2, 3, 3, 4, 1, 4, 5, 5, 5, 5, 6, 3, 2, 3, 6, 7, 7, 7, 7, 7, 7, 8, 2, 8, 1, 8, 2, 8, 9, 9, 1, 9, 9, 1, 9, 9, 10, 5, 10, 5, 2, 5, 10, 5, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 3, 4, 3, 12, 1, 12, 3, 4, 3, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Rémy Sigrist, Sep 25 2022

See A257522 for the corresponding denominators.

			Array A(n, k) begins:
  n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13
  ----+---------------------------------------------------
    1 |  2   3   4   5   6   7   8   9  10  11  12  13  14
    2 |  3   1   5   3   7   2   9   5  11   3  13   7  15
    3 |  4   5   2   7   8   1  10  11   4  13  14   5  16
    4 |  5   3   7   1   9   5  11   3  13   7  15   1  17
    5 |  6   7   8   9   2  11  12  13  14   3  16  17  18
    6 |  7   2   1   5  11   1  13   7   5   4  17   1  19
    7 |  8   9  10  11  12  13   2  15  16  17  18  19  20
    8 |  9   5  11   3  13   7  15   1  17   9  19   5  21
    9 | 10  11   4  13  14   5  16  17   2  19  20   7  22
   10 | 11   3  13   7   3   4  17   9  19   1  21  11  23
   11 | 12  13  14  15  16  17  18  19  20  21   2  23  24
   12 | 13   7   5   1  17   1  19   5   7  11  23   1  25
   13 | 14  15  16  17  18  19  20  21  22  23  24  25   2

Rémy Sigrist, Table of n, a(n) for n = 1..10011

Cf. A000034, A257522.

Flatten[Table[Numerator[1/i + 1/(m - i)], {m, 13}, {i, m - 1}]]

PARI
```
A(n,k) = numerator(1/n + 1/k)
```

A(n, k) = (n + k) / gcd(n + k, n*k).
A(n, k) = A(k, n).
A(n, 1) = n + 1.
A(n, n) = A000034(n).
A(n, n+1) = 2*n + 1.

cp's OEIS Frontend

A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

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